$$\text{My Favorite Creations :)}$$

$$\bbox[13px,#fdfdf9,border:5px dotted #7b5c7d ]{ \int_{0}^{2\pi}\left|\sin\left(\left(n-1\right)x-\frac{\pi}{2n}\right)\cos\left(nx\right)\right|dx = \frac{4n}{2n-1}\csc\left(\frac{\pi}{2n}\right)\sin\left(\frac{\pi}{n}\left\lceil \frac{n}{2}-1\right\rceil+\frac{\pi}{2n}\right)-\frac{4n-4}{2n-1}\csc\left(\frac{\pi}{2n-2}\right)\sin\left(\frac{\pi}{n-1}\left\lceil\frac{n}{2}-1\right\rceil\right)}$$

$$\bbox[13px,#f1fbef,border:5px outset #169602]{\int_{0}^{8}\frac{\arctan\left(\frac{\sqrt{64+3x^{2}}-8}{\sqrt{3}x}\right)}{\frac{5}{64}x^{2}-\frac{x}{4}+1}dx = \frac{\pi^{2}}{6}}$$

$$\bbox[13px,#f1ffff,border:5px double #a300ab ]{\int_{-\frac{1}{2}}^{0}\frac{\operatorname{arccot}\left(x+13\right)}{3x^{2}+3x+10}dx=\frac{1}{\sqrt{111}}\arctan\left(\sqrt{\frac{3}{37}}\right)\arctan\left(\frac{3}{19}\right)}$$

$$\bbox[0px,#0bddcd,border:3px solid #06897f]{\bbox[10px,#ffffff,border:3.3px inset #0bddcd]{\int_{0}^{\frac{\pi}{4}}\frac{\arctan\left(3\sqrt{2}\tan^{2}x-3\sqrt{2\tan^{2}x+1}\tan x+\sqrt{2}\right)\tan x}{\sqrt{2\tan^{2}x+1}}dx = \frac{\pi^{2}}{288}+\frac{\pi}{12}\arctan\left(4\sqrt{2}-3\sqrt{3}\right) }}$$

$$\bbox[13px,#fffbe9,border:5px inset #670000]{\int_{0}^{1}\frac{1}{3+x^{2}}\operatorname{arccot}\left(\sqrt{\frac{24\left(x+1\right)}{\left(x+3\right)^{2}}-1}\right)dx = \frac{\sqrt{3}\pi^{2}}{90}}$$

$$\bbox[10px,#e8dcdc,border:5px solid #88006a]{\bbox[13px,#ffffff,border:2px solid #88006a]{\int_{0}^{4}\frac{1}{\frac{5}{16}x^{2}-\frac{x}{2}+1}\arcsin\left(\frac{32}{3x^{2}+16}-1\right)dx=\frac{\pi^{2}}{6}}}$$

$$\bbox[px,border-left:4px dashed #120000]{\bbox[0px,#ffffff,border-right:4px dashed#120000]{\bbox[0px,border-top:8px ridge#670000]{\bbox[10px,#ffffff,border-bottom:8px groove#670000]{\int_{-\infty}^{0}\cos^{-1}\left(e^{\frac{x}{2}}\right)\cosh^{-1}\left(\frac{2e^{x}\left(1+e^{x}+\sqrt{1+e^{2x}}\right)^{2}}{1+\sqrt{1+e^{2x}}}+1\right)\tan^{-1}\left(\frac{e^{\frac{x}{2}}\left(1+e^{2x}\right)^{\frac{1}{4}}}{\sqrt{1+\sqrt{1+e^{2x}}+e^{2x}}}\right)dx=\frac{\pi^{4}}{64}}}}}$$

"Your eyes are as bright and honest as ever. You keep your head up, no matter what. I must've always been afraid... afraid you and I wouldn't wish for the same reality." ~Dr. Maruki~

(1) Email: codymaniquiz (at) gmail.com

(2) Discord: codyriceandothers

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#lgbtq+ he/him/his

Accelerator (方通行アクセラレータ: Ippō Tsūkō (Akuserarēta), lit. "One-Way Road") is the anti-hero in the A Certain Magical Index (Toaru Majutsu no Index) series. He is the 1st-ranked Level 5 and the strongest esper currently residing in Academy City.